Server Waiting Times in Infinite Supply Polling Systems With Preparation Times

Abstract

We consider a system consisting of a single server serving a fixed number of stations. At each station, there is an infinite queue of customers that have to undergo a preparation phase before being served. This model is connected to layered queueing networks, to an extension of polling systems and surprisingly to random graphs. We are interested in the waiting time of the server. For the case where the server polls the stations cyclically, we give a sufficient condition for the existence of a limiting waiting-time distribution and we study the tail behaviour of the stationary waiting time. Furthermore, assuming that preparation times are exponentially distributed, we describe in depth the resulting Markov chain. We also investigate a model variation where the server does not necessarily poll the stations in a cyclic order, but always serves the customer with the earliest completed preparation phase. We show that the mean waiting time under this dynamic allocation never exceeds that of the cyclic case, but that the waiting-time distributions corresponding to both cases are not necessarily stochastically ordered. Finally, we provide extensive numerical results investigating and comparing the effect of the system's parameters to the performance of the server for both models.

Throughput and Bottleneck Analysis of Tandem Queues With Nested Sessions

“…First, we designed three test suites with different number of sessions per server c = (c 1 , c 2 , c 3 ). Namely, one that represents a small system with c = (3, 2, 1), one for c = (12,8,5), and the third having c = (100, 90, 70). In all test suites we vary the values of the service rates μ 1 and μ 2 from 0.1 to 10 with a step size of 0.1.…”

“…The highest error that we found is 7.7% for the ten-layered network with the following parameters: c = (4, 3, 3, 2, 2, 2, 1, 1, 1, 1), and μ = (0.5, 0.75, 0.75, 0.9, 0.9, 0.9, 1, 1, 1, 1). Next to systems with the above stated number of sessions per server, we analyzed two more test suites with parameters c = (10,9,8,7,6,5,4,3,2,1) and c = (15,14,13,12,11,10,9,8,7,6). We varied the service rates in the range 20 − 500% of the following values: μ = (1,2,3,4,5,6,7,8,9,10) and μ = (1, 1, 1, 2, 2, 2, 2, 3, 3, 3) correspondingly.…”

“…Based on the above described test framework, we evaluate the Eff, SSR, and BStrength techniques on test suites, similar to those described in the previous section. Once again, for the three-layered systems we take: c = (3, 2, 1); c = (12,8,5), and c = (100, 90, 70) with μ 1 and μ 2 vary from 0.1 to 10 with a step size of 0.1. However, this time, next to the test cases where μ 3 is fixed at 1, we include such with μ 3 fixed at 10 and at 100.…”

“…Next to the above-mentioned approaches for generalized LQNs, there are a number of researches on special cases of such networks. A specific polling queueing system analyzed in Dorsman, Perel, and Vlasiou [5] can be considered as an instance of an LQN. Another example are systems with exactly two layers.…”